Advertisements
Advertisements
प्रश्न
Find the largest size of a rectangle that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter.
Advertisements
उत्तर
Let ABCD be the rectangle inscribed in a semicircle of radius 1 unit such that the vertices A and B lie on the diameter.
Let AB = DC = x and BC = AD = y.
Let O be the centre of the semicircle.
Join OC and OD. Then OC = OD = radius = 1.
Also, AD = BC and m∠A = m∠B = 90°.
∴ OA= OB
∴ OB = `(1)/(2) "AB" = x/(2)`
In right angled triangle OBC,
OB2 + BC2 = OC2
∴ `(x/2)^2 + y^2` = 12
∴ y2 = `1 - x^2/(4) = (1)/(4)(4 - x^2)`
∴ y = `(1)/(2)sqrt(4 - x^2)` ...[∵ y > 0]
Also of the triangle
= xy
= `x.(1)/(2)sqrt(4 - x^2)`
Let f(x) = `(1)/(2) xx sqrt(4 - x^2)`
= `(1)/(2)sqrt(4x2 - x^4)`
∴ f'(x) = `(1)/(2)d/dx(sqrt(4x^2 - x^4))`
= `(1)/(2) xx (1)/(2sqrt(4x^2 - x^4)) xx.d/dx(4x^2 - x^4)`
= `(1)/(4sqrt(4x^2 - x^4)) xx (4 xx 2x - 4x^3)`
= `(4x(2 - x2))/(4xsqrt(4 - x^2)`
= `(2 - x^2)/sqrt(4 - x^2)` ...[∵ x ≠ 0]
and
f"(x) = `d/dx((2 - x^2)/sqrt(4 - x^2))`
= `d/dx[((4 - x^2) - 2)/sqrt(4 - x^2)]`
= `d/dx[sqrt(4 - x^2) - (2)/sqrt(4 - x^2)]`
= `d/dx(sqrt(4 - x^2)) - 2d/dx(4 - x^2)^(-1/2)`
= `(1)/(2sqrt(4 - x^2)).d/dx(4- x^2) - 2(-1/2)(4 - x^2)^(-3/2).d/dx(4 - x^2)`
= `(1)/(2sqrt(4 -+ x^2)) xx (0 - 2x) + (1)/(4 - x^2)^(3/2) xx (0 - 2x)`
= `-x/sqrt(4 - x^2) - (2x)/(4 - x^2)^(3/2)`
= `(-x(4 - x^2) - 2x)/(4 - x^2)^(3/2)`
= `(-4x + x^3 - 2x)/(4 - x^2)^(3/2)`
= `(x^3 - 6x)/(4 - x^2)^(3/2)`
For maximum value of f(x),f'(x) = 0
∴ `(2 - x^2)/sqrt(4 - x^2)` = 0
∴ 2 – x2 = 0
∴ x2 = 2
∴ x = `sqrt(2)` ...[∵ x > 0]
Now, f"`(sqrt(2)) = ((sqrt(2))^3 - 6sqrt(2))/[4 - (sqrt(2))^2]^(3/2)`
= `(-4sqrt(2))/(2sqrt(2))`
= – 2 < 0
∴ by the second derivative test, f is maximum when x = `sqrt(2)`
When `x = sqrt(2), y = (1)/(2)sqrt(4 - x^2)`
= `(1)/(2)sqrt(4 - 2)`
= `(1)/(2) xx sqrt(2)`
= `(1)/sqrt(2)`
∴ `x = sqrt(2) and y = (1)/sqrt(2)`
Hence, the area of the rectangle is maximum (i.e. rectangle has the largest size) when its length is `sqrt(2)` units and breadth is `(1)/sqrt(2)`unit.
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
Find the maximum and minimum value, if any, of the following function given by f(x) = (2x − 1)2 + 3.
Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 − 72x − 18x2.
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
A square piece of tin of side 18 cm is to made into a box without a top by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
Find the absolute maximum and minimum values of the function f given by f (x) = cos2 x + sin x, x ∈ [0, π].
A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the maximum and minimum of the following functions : f(x) = `x^2 + (16)/x^2`
Find the maximum and minimum of the following functions : f(x) = x log x
Divide the number 20 into two parts such that sum of their squares is minimum.
The perimeter of a triangle is 10 cm. If one of the side is 4 cm. What are the other two sides of the triangle for its maximum area?
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
A wire of length 120 cm is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum
Max value of z equals 3x + 2y subject to x + y ≤ 3, x ≤ 2, -2x + y ≤ 1, x ≥ 0, y ≥ 0 is ______
If f(x) = 3x3 - 9x2 - 27x + 15, then the maximum value of f(x) is _______.
The sum of two non-zero numbers is 6. The minimum value of the sum of their reciprocals is ______.
If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
The maximum value of sin x . cos x is ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
If y = x3 + x2 + x + 1, then y ____________.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
Read the following passage and answer the questions given below.
|
|
- Is the function differentiable in the interval (0, 12)? Justify your answer.
- If 6 is the critical point of the function, then find the value of the constant m.
- Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute ‘maximum/absolute minimum values of the function.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.
Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is ______.
Let f(x) = (x – a)ng(x) , where g(n)(a) ≠ 0; n = 0, 1, 2, 3.... then ______.
The set of values of p for which the points of extremum of the function f(x) = x3 – 3px2 + 3(p2 – 1)x + 1 lie in the interval (–2, 4), is ______.
A cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
Find two numbers whose sum is 15 and when the square of one number multiplied by the cube of the other is maximum.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
For a function defined on an interval \[I\], which condition means that \[f\] has a maximum value at \[c\in I\]?
For a function defined on an interval \[I\], which condition means that \[f\] has a minimum value at \[c\in I\]?
A maximum or minimum value of \[f\] is called a(n) _____.
